Microscopic Description of the Breathing Mode and Nuclear Compressibility

1. Microscopic Description of the Breathing Mode and Nuclear Compressibility Presented By: David Carson Fuls Cyclotron REU Program 2005 Mentor: Dr. Shalom Shlomo

2. Introduction We use the microscopic Hartree-Fock (HF) based Random-Phase-Approximation (RPA) theory to describe the breathing mode in the 90Zr, 116Sn, 144Sm, and 208Pb nuclei, which are very sensitive to the nuclear matter incompressibility coefficient K. The value of K is directly related to the curvature of the equation of state, which is a very important quantity in the study of properties of nuclear matter, heavy ion collisions, neutron stars, and supernova. We present results of fully self-consistent HF+RPA calculations for the centroid energies of the breathing modes in the four nuclei using several Skyrme type nucleon-nucleon (NN) interactions and compare the results with available experimental data to deduce a value for K.

3. Nuclear Matter Incompressibility The value of K is directly related to the second derivative of the equation of state (EOS) of symmetric nuclear matter. Once we know that a two-body interaction is successful in determining the centroid energy of the monopole resonance, we can use that interaction to find the EOS and from that we can find the value of K. ρ = 0.16 fm-3 E/A [MeV] E/A = -16 MeV ρ [fm-3]

4. Classical Picture of the Breathing Mode In the classical description of the breathing mode, the nucleus is modeled after a drop of liquid that oscillates by expanding and contracting about its spherical shape. We consider the isoscalar breathing mode in which the neutrons and protons move in phase (∆T=0, ∆S=0).

5. In the scaling model, we have the matter density oscillates as We consider small oscillations, so є is very close to zero (≤ 0.1). Performing a Taylor expansion of the density we obtain,

6. We have, Where is equal to This nicely agrees with the transition density obtained from RPA calculations.

7. Microscopic Description of the Breathing Mode Ground State The ground state of the nucleus with A nucleons is given by an antisymmetric wave function which is, in the mean-field approximation, given by a Slater determinant. In the spherical case, the single-particle wave function is given in terms of the radial , the spherical spin harmonic , and the isospin functions:

8. The total Hamiltonian of the nucleus is written as a sum of the kinetic T and potential V energies Where The total energy E

9. Now we apply the variation principle to derive the Hartree-Fock equations. We minimize by varying with the constraint of particle number conservation, and obtain the Hartree-Fock equations

10. For the two-body nuclear potential Vij, we take a Skyrme type effective NN interaction given by, The Skyrme interaction parameters (ti, xi,α, and Wo) are obtained by fitting the HF results to the experimental data. This interaction is written in terms of delta functions which make the integrals in the HF equations easier to carry out.

11. For a spherical case the HF equations can be reduced to, where the effective mass , the central potential , and the spin-orbit potential are written in terms of the Skyrme parameters, matter density, charge density, and current density.

12. With an initial guess of the single-particle wave functions (usually the harmonic oscillator wave functions because they are known analytically) we can find the matter density, kinetic density, current density, and charge density. Once we know these values, we can use them to find the effective mass, central potential, and the spin- orbit potential. We then use these functions in the HF equations to find the new radial wave functions. We repeat the whole procedure with these new wave functions until convergence is reached. Method of Solving the HF Equations

13. Single-Particle Energies (in MeV) for 40Ca *TAMU Skyrme Interaction: B. K. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C 72, 014310 (2005).

14. Giant Resonance In HF based RPA theory, giant resonances are described as coherent superpositions of particle hole excitations of the ground state. In the Green’s Function formulation of RPA, one starts with the RPA-Green’s function which is given by where Vphis the particle-hole interaction and the free particle-hole Green’s function is defined as, where is the single-particle wave function, єi is the single-particle energy, and ho is the single-particle Hamiltonian.

15. We use the scattering operator F where for monopole excitation, to obtain the strength function and the transition density.

16. A Note on Self-Consistency In numerical implementation of HF based RPA theory, it is the job of the theorist to limit the numerical errors so that these are lower than the experimental errors. Some available HF+RPA calculations omit parts of the particle-hole interaction that are numerically difficult to implement, such as the spin-orbit or Coulomb parts. Omission of these terms leads to self-consistency violation, and the shift in the centroid energy can be on the order of 1 MeV or 5 times the experimental error. The calculations we have carried out are fully self-consistent. Note: For example: E = 14 MeV (in 208Pb), and K = 230 MeV, then a ΔE = 1 MeV leads to ΔK = 35 MeV.

17. Isoscalar Monopole Strength Functions 90Zr 116Sn S(E) [fm4/MeV] 144Sm 208Pb E [MeV]

18. Fully Self-Consistent HF Based RPA Results For Breathing Mode Energy (in MeV) • TAMU Data: D. H. Youngblood et al, Phys. Rev. C 69, 034315 (2004); C 69, 054312(2004). • Nguyen Van Giai and H. Sagawa, Phys. Lett. B106, 379 (1981). • c) TAMU Interaction: B. K. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C 72, 014310 (2005).

19. Conclusion After doing the fully self-consistent HF+RPA calculations for the centroid energy of the breathing mode in the four nuclei, using the two Skyrme interactions SG2 and KDE0, we have deduced a value of K = 230 +/- 20 MeV.

20. Work done at: Acknowledgments

21. Work supported by: Grant numbers: PHY-0355200 PHY-463291-00001 Grant number: DOE-FG03-93ER40773